text: During its life, a plant or animal is in equilibrium with its surroundings by exchanging carbon either with the atmosphere or through its diet. It will, therefore, have the same proportion of  as the atmosphere, or in the case of marine animals or plants, with the ocean. Once it dies, it ceases to acquire , but the  within its biological material at that time will continue to decay, and so the ratio of  to  in its remains will gradually decrease. Because  decays at a known rate, the proportion of radiocarbon can be used to determine how long it has been since a given sample stopped exchanging carbon – the older the sample, the less  will be left.
  The equation governing the decay of a radioactive isotope is:
  where N is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. λ is a constant that depends on the particular isotope; for a given isotope it is equal to the reciprocal of the mean-life – i.e. the average or expected time a given atom will survive before undergoing radioactive decay. The mean-life, denoted by τ, of  is 8,267 years, so the equation above can be rewritten as:
  The sample is assumed to have originally had the same / ratio as the ratio in the atmosphere, and since the size of the sample is known, the total number of atoms in the sample can be calculated, yielding N, the number of  atoms in the original sample. Measurement of N, the number of  atoms currently in the sample, allows the calculation of t, the age of the sample, using the equation above.
  The half-life of a radioactive isotope (usually denoted by t) is a more familiar concept than the mean-life, so although the equations above are expressed in terms of the mean-life, it is more usual to quote the value of 's half-life than its mean-life. The currently accepted value for the half-life of  is 5,700 ± 30 years. This means that after 5,700 years, only half of the initial  will remain; a quarter will remain after 11,400 years; an eighth after 17,100 years; and so on.
  The above calculations make several assumptions, such as that the level of  in the atmosphere has remained constant over time. In fact, the level of  in the atmosphere has varied significantly and as a result, the values provided by the equation above have to be corrected by using data from other sources. This is done by calibration curves (discussed below), which convert a measurement of  in a sample into an estimated calendar age. The calculations involve several steps and include an intermediate value called the "radiocarbon age", which is the age in "radiocarbon years" of the sample: an age quoted in radiocarbon years means that no calibration curve has been used − the calculations for radiocarbon years assume that the atmospheric / ratio has not changed over time.
  Calculating radiocarbon ages also requires the value of the half-life for . In Libby's 1949 paper he used a value of 5720 ± 47 years, based on research by Engelkemeir et al. This was remarkably close to the modern value, but shortly afterwards the accepted value was revised to 5568 ± 30 years, and this value was in use for more than a decade. It was revised again in the early 1960s to 5,730 ± 40 years, which meant that many calculated dates in papers published prior to this were incorrect (the error in the half-life is about 3%). For consistency with these early papers, it was agreed at the 1962 Radiocarbon Conference in Cambridge (UK) to use the "Libby half-life" of 5568 years. Radiocarbon ages are still calculated using this half-life, and are known as "Conventional Radiocarbon Age". Since the calibration curve (IntCal) also reports past atmospheric  concentration using this conventional age, any conventional ages calibrated against the IntCal curve will produce a correct calibrated age. When a date is quoted, the reader should be aware that if it is an uncalibrated date (a term used for dates given in radiocarbon years) it may differ substantially from the best estimate of the actual calendar date, both because it uses the wrong value for the half-life of , and because no correction (calibration) has been applied for the historical variation of  in the atmosphere over time.